Thirdly, unlike the smooth approximate rarefaction wave, the sign of \( \bar{u}_x \)  is not determined for viscous contact wave.
We have to estimate the terms including the weight \( \bar{u}_x \)  such as \( \int _0^t\int _{\mathbb {R}} \bar{u}_x \left( \varepsilon E^2 + b^2 \right) \,\mathrm {d}x\mathrm {d}\tau \)  (see in (REF )), etc.
To overcome this difficulty, we observe that the fifth equation of \((\ref {raodong}) \)  can be transformed as \( \left( vb \right)_t - \left( E + \psi b + \bar{u} b \right)_x = 0 \) ,
which contains the derivative of the compound dissipative term \(E + \psi b + \bar{u} b\) .
This transformation of \((\ref {raodong})_5 \)  is quite different from the structure of \( (\ref {yuanfangcheng})_5 \)  in the Eulerian coordinates.
Thanks to this key observation, by using the heat kernel type of inequality in Lemma REF  borrowed from [1]} and then employing the previous compound time-space integrable good term \(\int _0^t\int _\mathbb {R} v(E+\psi b+\bar{u} b)^2\mathrm {d}x{\rm {d}}\tau \) ,
we can obtain the desired estimates; see the proof of (REF ) in Lemma REF .
